Oscillations (SHM) · JEE Main previous year questions

A car moving with a speed of $54km/h$ takes a turn of radius $20\mathrm{m}$. A simple pendulum is suspended from the ceiling of the car. Determine the angle made by the string of the pendulum with the vertical during the turning. (Take $g=10m/{\mathrm{s}}^2$)

1. A.${\tan }^{-1}(0.5)$
2. B.${\tan }^{-1}(0.75)$
3. C.${\tan }^{-1}(1.125)$
4. D.${\tan }^{-1}(0.25)$

The frequency of oscillation of a mass $m$ suspended by a spring is $v_1$. If the length of the spring is cut to half, the same mass oscillates with frequency $v_2$. The value of ${}^{v_2}/{}_{v_1}$ is ____ .

1. A.$1$
2. B.$2$
3. C.$\sqrt{2}$
4. D.$\sqrt{3}$

A particle is executing simple harmonic motion. Its amplitude is $A$ and time period is $5\sec$. The time required by it to move from $x=A$ to $x=\frac{A}{\sqrt{2}}$ is ____ $\sec$.

1. A.1/4
2. B.5/4
3. C.5/8
4. D.3/8

A spring stretches by $2\text{ mm}$ when it is loaded with a mass of $200\text{ g}$. From equilibrium position the mass is further pulled down by $2\text{ mm}$ and released. The frequency associated with the system and maxmimum energy in the spring are ________ $\text{Hz}$ and ________ $\text{J}$, respectively. (Take $g=10{\text{ m/s}}^2$)

1. A.$\frac{5\sqrt{50}}{\pi }$ and $8\times 10^{-3}$
2. B.$\frac{5\sqrt{50}}{\pi }$ and $8$
3. C.$10\sqrt{50}$ and $2\times 10^{-3}$
4. D.$\frac{5\sqrt{50}}{\pi }$ and $16\times 10^{-3}$

Match List - I with List - II. $$\begin{array}{ll}\text{List - I}& \text{List - II}\\ \text{A. }{\sin }^2\omega t& \text{I. Periodic with time period }T=\frac{\pi }{\omega }\text{ but not simple harmonic motion (SHM)}\\ \text{B. }{\sin }^3(2\omega t)& \text{II. Periodic with time period }T=\frac{2\pi }{\omega }\text{ but Not SHM}\\ \text{C. }\sin (\omega t)+\cos (\pi \omega t)& \text{III. Periodic with time period }T=\frac{\pi }{\omega }\text{ and SHM}\\ \text{D. }\cos \omega t+\cos 2\omega t& \text{IV. Non-periodic}\end{array}$$ Choose the correct answer from the options given below :

1. A.A-III, B-I, C-IV, D-II
2. B.A-II, B-I, C-III, D-IV
3. C.A-III, B-II, C-IV, D-I
4. D.A-II, B-I, C-IV, D-III

The velocity of a particle executing simple harmonic motion along $x$-axis is described as $v^2=50-x^2$, where $x$ represents displacement. If the time period of motion is $\frac{x}{7}\text{ s}$, the value of $x$ is ________.

A rigid dipole undergoes a simple harmonic motion about its centre in the presence of an electric field ${\vec{\mathrm{E}}}_1={\mathrm{E}}_0\hat{x}$. If another electric field ${\vec{\mathrm{E}}}_2=2{\mathrm{E}}_0(\hat{y}+\hat{z})$ is introduced to the system, what will be the percentage change in the frequency of the oscillation (approximate)?

1. A.$73\%$
2. B.$63\%$
3. C.$83\%$
4. D.$53\%$

The displacement of a particle, executing simple harmonic motion with time period $T$, is expressed as $x(t)=A\sin \omega t$, where $A$ is the amplitude. The maximum value of potential energy of this oscillator is found at $t=T/2\beta$. The value of $\beta$ is ____ .

The time period of a simple harmonic oscillator is $T=2\pi \sqrt{\frac{k}{m}}$. Measured value of mass ($m$) of the object is 10 g with an accuracy of 10 mg and time for 50 oscillations of the spring is found to be 60 s using a watch of 2 s resolution. Percentage error in determination of spring constant ($k$) is ________%.

1. A.3.43
2. B.7.60
3. C.3.35
4. D.6.76

A spring of force constant $15N/m$ is cut into two pieces. If the ratio of their length is $1:3$, then the force constant of smaller piece is ____ N/m.

1. A.20
2. B.45
3. C.60
4. D.15

A simple pendulum of string length 30 cm performs 20 oscillations in 10 s . The length of the string required for the pendulum to perform 40 oscillations in the same time duration is ____ cm . [Assume that the mass of the pendulum remains same.]

1. A.0.75
2. B.7.5
3. C.15
4. D.120

A simple pendulum made of mass 10 g and a metallic wire of length 10 cm is suspended vertically in a uniform magnetic field of 2 T . The magnetic field direction is perpendicular to the plane of oscillations of the pendulum. If the pendulum is released from an angle of $60^{\circ }$ with vertical, then maximum induced EMF between the point of suspension and point of oscillation is ____ mV . (Take $g=10{\text{ m/s}}^2$ )

The kinetic energy of a simple harmonic oscillator is oscillating with angular frequency of 176 rad/s. The frequency of this simple harmonic oscillator is ____ Hz. [ take $\pi =\frac{22}{7}$ ]

1. A.88
2. B.14
3. C.28
4. D.176

Two simple pendulums having lengths $l_1$ and $l_2$ with negligible string mass undergo angular displacements ${\theta }_1$ and ${\theta }_2$, from their mean positions, respectively. If the angular accelerations of both pendulums are same, then which expression is correct?

1. A.${\theta }_1{l}_2={\theta }_2{l}_1$
2. B.${\theta }_1{l}_1={\theta }_2{l}_2$
3. C.${\theta }_1{l}_2^2={\theta }_2{l}_1^2$
4. D.${\theta }_1{l}_1^2={\theta }_2{l}_2^2$

1. A.B, C, D Only
2. B.C, D, E Only
3. C.A, B, D Only
4. D.A, B, C Only

A particle is subjected to two simple harmonic motions as : $x_1=\sqrt{7}\sin 5t\text{ cm}$ and $x_2=2\sqrt{7}\sin \left(5t+\frac{\pi }{3}\right)\text{cm}$ where $x$ is displacement and $t$ is time in seconds. The maximum acceleration of the particle is $x\times 10^{-2}{\text{ ms}}^{-2}$. The value of $x$ is :

1. A.$5\sqrt{7}$
2. B.125
3. C.$25\sqrt{7}$
4. D.175

In a moving coil galvanometer, two moving coils ${\mathrm{M}}_1$ and ${\mathrm{M}}_2$ have the following particulars : ${\mathrm{R}}_1=5\Omega$, ${\mathrm{N}}_1=15$, ${\mathrm{A}}_1=3.6\times 10^{-3}{\mathrm{m}}^2$, ${\mathrm{B}}_1=0.25\mathrm{T}$ ${\mathrm{R}}_2=7\Omega$, ${\mathrm{N}}_2=21$, ${\mathrm{A}}_2=1.8\times 10^{-3}{\mathrm{m}}^2$, ${\mathrm{B}}_2=0.50\mathrm{T}$ Assuming that torsional constant of the springs are same for both coils, what will be the ratio of voltage sensitivity of $M_1$ and $M_2$ ?

1. A.$1:1$
2. B.$1:3$
3. C.$1:2$
4. D.$1:4$

Two bodies A and B of equal mass are suspended from two massless springs of spring constant $k_1$ and $k_2$, respectively. If the bodies oscillate vertically such that their amplitudes are equal, the ratio of the maximum velocity of A to the maximum velocity of B is

1. A.$\sqrt{\frac{k_2}{k_1}}$
2. B.$\sqrt{\frac{k_1}{k_2}}$
3. C.$\frac{k_2}{k_1}$
4. D.$\frac{k_1}{k_2}$

A particle is executing simple harmonic motion with time period $2\mathrm{s}$ and amplitude $1\mathrm{cm}$ . If $\mathrm{D}$ and $\mathrm{d}$ are the total distance and displacement covered by the particle in $12.5\mathrm{s}$ , then $\frac{\mathrm{D}}{\mathrm{d}}$ is

1. A.$10$
2. B.$\frac{16}{5}$
3. C.$25$
4. D.$\frac{15}{4}$

A massless spring gets elongated by amount $x_1$ under a tension of $5\text{ N}$ . Its elongation is $x_2$ under the tension of $7\text{ N}$ . For the elongation of $(5x_1-2x_2)$, the tension in the spring will be,

1. A.$20\text{ N}$
2. B.$39\text{ N}$
3. C.$11\text{ N}$
4. D.$15\text{ N}$

The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes of $4\text{ m}$, $2{\text{ ms}}^{-1}$ and $16{\text{ ms}}^{-2}$ at a certain instant. The amplitude of the motion is $\sqrt{x}\text{ m}$ where $x$ is ______.

A particle of mass $0.50\mathrm{kg}$ executes simple harmonic motion under force $F=-50({\mathrm{Nm}}^{-1})x$. The time period of oscillation is $\frac{x}{35}s$. The value of $x$ is ____. (Given $\pi =\frac{22}{7}$)

An object of mass $0.2\mathrm{kg}$ executes simple harmonic motion along $x$ axis with frequency of $\left(\frac{25}{\pi }\right)\mathrm{Hz}$. At the position $x=0.04\mathrm{m}$ the object has kinetic energy $0.5\mathrm{J}$ and potential energy $0.4\mathrm{J}$. The amplitude of oscillation is ____ $\mathrm{cm}$.

An elastic spring under tension of $3\mathrm{N}$ has a length $a$. Its length is $b$ under tension $2\mathrm{N}$. For its length $(3a-2b)$, the value of tension will be ____ $\mathrm{N}$.

1. A.$E/\sqrt{2}$
2. B.$2E$
3. C.$E\sqrt{2}$
4. D.$E$